On the groups of periodic links

TL;DR

This paper establishes a relationship between the fundamental groups of periodic links and their quotient links by analyzing homomorphisms to finite groups, revealing algebraic invariants preserved under periodicity.

Contribution

It introduces a method to relate the groups of periodic links to their quotient links via homomorphism counts to finite groups with order not divisible by the period prime.

Findings

Groups of periodic links can be studied through homomorphism counts.

The relation holds for any finite group whose order is coprime with the prime period.

Provides a new algebraic approach to understanding periodic links.

Abstract

It is shown that, if a link $\tilde{L}\subset S^3$ is $p^k$-periodic with $p$ prime and $k\ge 1$, and $L$ is the quotient link, then the groups of $\tilde{L}$ and $L$ can be related by counting homomorphisms to any finite group $\Gamma$ whose order is not divisible by $p$.